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Difference between revisions of "2004 AIME I Problems/Problem 1"

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== Solution 2==
 
== Solution 2==
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There are only <math>7</math> possible values of <math>n</math>: <math>3210, 4321, 5432, \cdots, 9876</math>. <math>3210</math> gives a remainder of <math>28</math> when divided by <math>37</math>. To calculate the remainders of the other integers, notice that each number is <math>1111</math> more than the previous number. Since <math>1111</math> gives a remainder of <math>1</math> when divided by <math>37</math>, the remainders of the other integers will be <math>29, 30, 31, \cdots, 34</math>. Therefore, our answer is <math>28 + 29 + 30 + 31 + 32 + 33 + 34 = \frac{28 + 34}{2} \cdot 7 = \boxed{217}</math>.
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There are only <math>7</math> possible values of <math>n</math>: <math>3210, 4321, 5432, \cdots, 9876</math>.  
 +
 
 +
<math>3210</math> gives a remainder of <math>28</math> when divided by <math>37</math>. To calculate the remainders of the other integers, notice that each number is <math>1111</math> more than the previous number. Since <math>1111</math> gives a remainder of <math>1</math> when divided by <math>37</math>, the remainders of the other integers will be <math>29, 30, 31, \cdots, 34</math>.  
 +
 
 +
Therefore, our answer is <math>28 + 29 + 30 + 31 + 32 + 33 + 34 = \frac{28 + 34}{2} \cdot 7 = \boxed{217}</math>.
 
~Viliciri
 
~Viliciri
  

Latest revision as of 22:56, 2 December 2024

Problem

The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?

Solution 1

A brute-force solution to this question is fairly quick, but we'll try something slightly more clever: our numbers have the form ${\underline{(n+3)}}\,{\underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}}$$= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n$, for $n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace$.

Now, note that $3\cdot 37 = 111$ so $30 \cdot 37 = 1110$, and $90 \cdot 37 = 3330$ so $87 \cdot 37 = 3219$. So the remainders are all congruent to $n - 9 \pmod{37}$. However, these numbers are negative for our choices of $n$, so in fact the remainders must equal $n + 28$.

Adding these numbers up, we get $(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}$

Solution 2

There are only $7$ possible values of $n$: $3210, 4321, 5432, \cdots, 9876$.

$3210$ gives a remainder of $28$ when divided by $37$. To calculate the remainders of the other integers, notice that each number is $1111$ more than the previous number. Since $1111$ gives a remainder of $1$ when divided by $37$, the remainders of the other integers will be $29, 30, 31, \cdots, 34$.

Therefore, our answer is $28 + 29 + 30 + 31 + 32 + 33 + 34 = \frac{28 + 34}{2} \cdot 7 = \boxed{217}$. ~Viliciri

See also

2004 AIME I (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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